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Theorem raleqbidvv 3334
Description: Version of raleqbidv 3346 with additional disjoint variable conditions, not requiring ax-8 2110 nor df-clel 2816. (Contributed by BJ, 22-Sep-2024.)
Hypotheses
Ref Expression
raleqbidvv.1 (𝜑𝐴 = 𝐵)
raleqbidvv.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
raleqbidvv (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidvv
StepHypRef Expression
1 raleqbidvv.1 . 2 (𝜑𝐴 = 𝐵)
2 raleqbidvv.2 . . 3 (𝜑 → (𝜓𝜒))
32adantr 480 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
41, 3raleqbidva 3332 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ral 3062  df-rex 3071
This theorem is referenced by:  rexeqbidvvOLD  3337  raleqbi1dv  3338  gpg5nbgrvtx03star  48036  postcposALT  49172
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